Quote from Royal Tunnel Wiki Page "At the end of each
path (with the exception of the Endless Path), you can find an egg
containing one of the Regi quintett. The Beginner Path gives
Regirock, the Advanced Path gives Regice, the Pro Path gives
Registeel and the Split-Decision Path gives Regieleki or
Regidrago.
At the Endless Path you will be able to receive some more Regi-eggs
randomly [a chance to get one every 50 levels], and every 1000,
2000, 3000 etc. level you will receive a random Shiny Regi-Egg!
Note: You need a free space in your party to be able to receive a
Regi-egg!
There is a small chance of receiving a random gem after correctly
answering a question and you can revive up to 100 gems each
day."
You can also obtain Regigigas through the Royal Tunnel, once you
have unlocked the Endless Path:
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Place Regirock, Registeel, Regice, and
Relicanth in you party, in that order, with rest of the slots
empty. You will obtain a Regigigas Egg. Shiny Regigigas can be
obtained by using a Shiny Regi Trio instead [Shiny Relicanth is not
required].
Necrozma quest spoilers:
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You can also find one of the Old Vinyl
Shards required as part of the Necrozma quest at the Royal Tunnel,
in a similar way to finding gems.